Distributed Path Tracking for Autonomous Underwater Vehicles Based on Pseudo Position Feedback

Abstract

In this paper, we consider the distributed polynomial path tracking problem for a swarm of autonomous underwater vehicles (AUVs) modeled by second-order uncertain multi-agent systems. The application scenario of this paper has three distinguished characteristics. First, the communication network for the multi-agent system is unreliable and switching. Under the jointly connected condition, the communication network can be disconnected the entire time. Second, it is supposed that only the relative position between AUVs can be obtained for trajectory tracking control. Third, the AUV dynamics are subject to uncertain system parameters. By applying the cooperative output regulation control framework, a novel distributed robust control scheme is proposed to solve the distributed path tracking problem, which consists of three parts. First, to cope with communication network uncertainty, the distributed observer was invoked to recover the polynomial path for each AUV. Second, based on the relative position measurement between AUVs, a pseudo position estimator was adopted to generate the pseudo position for each AUV. Finally, based on the estimated polynomial path and the pseudo position, a certainty equivalent robust internal model control law was synthesized to achieve asymptotic reference trajectory tracking, where the internal model compensator aims to tackle uncertain system parameters. Numerical simulations are provided to validate the effectiveness of the proposed control scheme.

1. Introduction

Over the past two decades, tremendous attention has been paid to the cooperative control of swarm systems, which has led to fruitful research results [1,2,3], and has found broad applications in the area of unmanned swarm systems, such as collaborative searching, escorting, localization and mapping by unmanned aerial vehicles [4,5,6] and unmanned mobile robots [7,8,9]. In particular, the motion in formation for unmanned swarm systems is the foundation of many tasks, where all of the individuals in the swarm are supposed to cooperate to maintain some desired relative position and/or orientation with respect to each other while following some common reference path. The existing control approaches dealing with the formation problem can be classified roughly into three categories [10]. The first category is the behavior-based approach, which was first proposed by Balch and Arkin in [11] for four kinds of formation, namely, line, column, wedge and diamond. The guidance and formation are fulfilled by designing and weighting motor schemas. Typical works on the behavior-based approach can be further found in [12,13]. The second category is the leader–follower approach, which defines the global formation by a series of pre-specified leader-following tracking problems. Since the classic leader-following tracking problem has been well treated in the literature, the leader–follower approach has become an effective way to solve the formation problem [14,15]. The third category is the virtual structure approach, where the formation is defined by the static or dynamic relative position/orientation of each agent with respect to a virtual leader. Recently, owing to the application of graph theory in the control of multi-agent systems, the virtual structure approach has become the most popular control strategy for distributed formation control over sparse and vulnerable communication networks [16].

The rapid development of autonomous underwater vehicles (AUVs) has greatly enabled underwater tasks, such as environment inspection, military surveillance, oceanographic observation, site searching and so on [17,18,19]. In contrast to a single AUV, AUV swarms show a better system reliability against failure and system adaptability for complex tasks [20,21,22]. There have thus far been extensive works devoted to the formation control of an AUV swarm. A group of torpedo-type AUVs was investigated in [23] under the centralized management by an unmanned surface vessel, constituting a star-like command and communication system, with the unmanned surface vessel being the center. The formation is achieved by assigning each AUV its corresponding reference trajectory. Both references [24,25] adopted the leader–follower control structure. Environmental disturbances and input saturation were considered in [24], tackled by dynamic surface control and a hyperbolic tangent function, respectively. To simultaneously deal with communication delay, packet discreteness and dropout, a curve fitting method was developed by [25] to make a prediction for the states of the AUV. On the other hand, the virtual structure approach based on graph theory was invoked in [26,27,28,29,30,31]. Li et al. considered prescribed performance control for underactuated AUVs subject to uncertain dynamics and disturbances [26]. By using radial basis functions, the lumped uncertainties of the entire system were transformed into a linearly parameterized form with a single uncertain parameter. Different from [26], model predictive control and an extended state observer were employed in [27] to cope with unknown ocean current disturbances. Time delay issues were discussed in [28,29]. Both bounded and unbounded communication delays were considered in [28], and two separate communication networks for both position and velocity were implemented to realize formation trajectory tracking. A consistent control algorithm was proposed by [29] to deal with communication delays by Gershgorin disk theorem and Nyquist law. To recover the information of the global trajectory, distributed observers were resorted to in [30,31]. A neural-network-based deterministic learning approach was established in [30] to tackle the nonlinear uncertain dynamics of the AUVs. To reduce the risk of actuator saturation, a hyperbolic tangent function was adopted in [31] to design a decentralized formation tracking control law that can also handle system nonlinearities and measurement noises.

From the perspective of the system structure, swarm systems can be categorized as a leaderless swarm and leader–follower swarm, where the formation control of swarms is mainly involved with the latter one by viewing the formation command as the leader. The cooperative output regulation theory proves to be an effective control framework tackling the distributed control of leader–follower swarm systems [32], which can also be viewed as an extension of the virtual structure approach. In the cooperative output regulation framework, the exosystem is viewed as the leader, and each agent is viewed as a follower. The control objectives are twofold: (1) the stability of the closed-loop system should be guaranteed; (2) the output of each agent should track a class of reference signals while rejecting a class of external disturbances. So far, there have been some attempts applying the cooperative output regulation theory to solve the distributed formation problem [33,34,35,36,37]. Wang considered a linear swarm system over a static communication network [33]. By taking the system matrix of the exosystem as prior knowledge, a distributed internal model was constructed for each agent utilizing virtual errors. Hua et al. studied the case where the virtual leader has an external input [34]. A sign-function-based distributed observer was proposed to recover the state of the virtual leader. Like [33], the system matrix of the virtual leader should be known in advance so as to obtain the solution to the regulator equations. The situation of multiple virtual leaders was considered in [35], and the formation problem was integrated with the objective of containment control. Through a thorough analysis on the Laplacian of the associated communication graph, an adaptive distributed control law was proposed to solve the formation problem. Similar to [33], Li et al. also made use of the virtual-error-based distributed internal model [36], while both the swarm system and the virtual leader considered in [36] are nonlinear. Local stability results were obtained by performing Jacobian linearization around the origin of the closed-loop system. In practice, the communication network for the swarm might not always be safe or reliable. Huang and Dong investigated the scenario of false data injection into the communication network [37]. By adopting linear matrix inequality techniques, a robust output regulation control law was proposed so that the tracking errors can be guaranteed to be uniformly ultimately bounded under uniform feedback quantization.

In this paper, we consider the distributed path tracking problem for a swarm of AUVs modeled by second-order multi-agent systems. The AUV swarm should track a class of polynomial path signals while keeping a desired relative position with respect to each other. Different from the existing results, the application scenario of this paper has three distinguished characteristics. First, the unreliable communication network for the multi-agent system was modeled as a switching graph satisfying the jointly connected condition, which allows the communication network topology to be disconnected the entire time. Second, in an underwater environment, absolute position measurement might not be feasible. To address this issue, in this paper, we consider the case where only the relative position between AUVs can be obtained for trajectory tracking control. Third, in the presence of uncertain mass, inertia and velocity damping, the AUV dynamics are assumed to contain uncertain system parameters. By applying the cooperative output regulation control framework, a novel distributed robust control scheme is proposed to solve the distributed path tracking problem, which consists of three parts. First, to cope with communication network uncertainty, the distributed observer was invoked to recover the polynomial path for each AUV, which decouples the dynamics of the AUVs and the virtual leader, thus making it easy to conceive a feasible distributed control scheme under the unreliable communication environment. Second, based on the relative position measurement between AUVs, a pseudo position estimator was adopted to generate the pseudo position for each AUV. The interesting fact regarding the pseudo position lies in that, though it will not converge to the absolute position of the AUV, the differences between the absolute positions and pseudo positions of all of the AUVs will converge to a common constant vector, which paves the way to the success of the proposed distributed control scheme. Finally, based on the estimated polynomial path and the pseudo position, a certainty equivalent robust internal model control law was synthesized to achieve asymptotic reference trajectory tracking, where the internal model compensator aims to tackle uncertain system parameters. It was rigorously proven that the distributed path tracking problem can be solved by the proposed distributed robust control scheme. In contrast to the existing results, the main contributions of this paper are twofold:

• In terms of communication topology, reference [23] adopted a centralized structure with the unmanned surface vessel as the communication center, whereas [24,25] adopted the leader–follower approach, whose communication topology is essentially a pure tree-like structure. In [26,27,28,29,30,31], based on graph theory, the communication topology is relaxed to be freely designed as long as the associated communication graph is connected. However, in all of these works [26,27,28,29,30,31], the communication topology should be connected the entire time, which might be impractical for certain application scenarios. In contrast, in this paper, we allow the communication network to be jointly connected, which can be disconnected for the entire time, thus greatly reducing the requirement imposed on the communication network.
• In  [23,24,25,26,27,28,29,30,31], the absolute position feedback of the AUV is necessary to stabilize the closed-loop system dynamics so that the control objective of tracking or formation can be fulfilled, while, as indicated by [18,20,22], the global localization of AUVs might be costly in an underwater environment, and might sometimes even be impossible, such as in a deep ocean environment. To address this issue, in this paper, the proposed control method only needs a relative position measurement of the neighboring AUVs over the communication network to achieve reference path tracking and maintain a relative formation, which makes it more practical and cost competitive in contrast to the existing results.

2. Graph Notation

A graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ is defined by a node set $\mathcal{V}=\{1,\dots ,N\}$ and an edge set $\mathcal{E}\subseteq \mathcal{V}\times \mathcal{V}$. For $i,j=1,2,\dots ,N$, $i\ne j$, $(i,j)\in \mathcal{E}$ means that there exists an edge in $\mathcal{E}$ from node i to node j. If $(i,j)\in \mathcal{E}$, then node i is called a neighbor of node j. Let ${\mathcal{N}}_i=\{j,(j,i)\in \mathcal{E}\}$ denote the neighbor set of node i. If $(i,j)\in \mathcal{E}$ if and only if $(j,i)\in \mathcal{E}$, then the edge $(i,j)$ is called undirected. If all of the edges of a graph are undirected, then the graph is called undirected. If $\mathcal{G}$ contains a set of edges of the form $(i_1,i_2),(i_2,i_3),\dots ,(i_k,i_{k+1})$, then the set $\{(i_1,i_2),(i_2,i_3),\dots ,(i_k,i_{k+1})\}$ is called a path of $\mathcal{G}$ from node $i_1$ to node $i_{k+1}$, and node $i_{k+1}$ is said to be reachable from node $i_1$. A graph $\mathcal{G}$ is said to contain a spanning tree if there exists a node in $\mathcal{G}$ such that all of the other nodes are reachable from it, and this node is called the root of the spanning tree. Given a set of m graphs ${\mathcal{G}}_k=(\mathcal{V},{\mathcal{E}}_k),k=1,\dots ,m$, the graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ with $\mathcal{E}={\bigcup }_{k=1}^m{\mathcal{E}}_k$ is called the union of ${\mathcal{G}}_k$, denoted by $\mathcal{G}={\bigcup }_{k=1}^m{\mathcal{G}}_k$.

A time signal $\sigma \left(t\right):[0,+\infty )\to \mathcal{M}=\{1,\dots ,m\}$ for some positive integer m is called a piecewise constant switching signal with dwell time $\tau$ for some $\tau >0$ if there exists a time sequence $\{t_k,k=0,1,2,\dots ,\}$ satisfying $t_0=0$; for any positive integer k, $t_k-t_{k-1}\ge \tau$; $\sigma \left(t\right)=p$, $p\in \mathcal{M}$, for all $t\in [t_{k-1},t_k)$. Given a node set $\mathcal{V}=\{1,\dots ,N\}$ and a piecewise constant switching signal $\sigma \left(t\right)$, define a switching graph ${\mathcal{G}}_{\sigma \left(t\right)}=(\mathcal{V},{\mathcal{E}}_{\sigma \left(t\right)})$ where ${\mathcal{E}}_{\sigma \left(t\right)}\subseteq \mathcal{V}\times \mathcal{V}$ for all $t\ge 0$. For a switching graph, let ${\mathcal{N}}_i\left(t\right)$ denote the neighbor set of node i at time instant t. Associated with a switching graph ${\mathcal{G}}_{\sigma \left(t\right)}$, the matrix ${\mathcal{A}}_{\sigma \left(t\right)}=\left[a_{ij}\left(t\right)\right]\in {\mathbb{R}}^{N\times N}$ is called a time-varying weighted adjacency matrix of ${\mathcal{G}}_{\sigma \left(t\right)}$ if $a_{ii}\left(t\right)=0$; $a_{ij}\left(t\right)>0\iff (j,i)\in {\mathcal{E}}_{\sigma \left(t\right)}$; and $a_{ij}\left(t\right)=0$ otherwise. Let ${\mathcal{L}}_{\sigma \left(t\right)}=\left[l_{ij}\left(t\right)\right]\in {\mathbb{R}}^{N\times N}$ be such that $l_{ii}\left(t\right)={\sum }_{j=1}^N{a}_{ij}\left(t\right)$ and $l_{ij}\left(t\right)=-a_{ij}\left(t\right)$ if $i\ne j$. Then, ${\mathcal{L}}_{\sigma \left(t\right)}$ is called the Laplacian of ${\mathcal{G}}_{\sigma \left(t\right)}$ associated with ${\mathcal{A}}_{\sigma \left(t\right)}$.

3. Problem Statement

In this paper, we consider the distributed polynomial path tracking control problem for an AUV swarm system consisting of N AUVs. As in [38], we suppose the AUV is of full actuation. Then, for $i=1,\dots ,N$, the dynamics of the ith AUV take the following form

$$\begin{array}{cc}\hfill {\dot{p}}_i& =v_i\hfill \end{array}$$

(1a)

$$\begin{array}{cc}\hfill {\dot{v}}_i& ={\gamma }_{i1}{p}_i+{\gamma }_{i2}{v}_i+{\gamma }_{i3}{u}_i\hfill \end{array}$$

(1b)

where $p_i,v_i,u_i\in {\mathbb{R}}^n$ denote the generalized position, velocity and control input of the ith AUV, respectively. ${\gamma }_{i1},{\gamma }_{i2},{\gamma }_{i3}\in \mathbb{R}$ are unknown system parameters satisfying ${\gamma }_{ij}={\gamma }_{ij}^o+\Delta {\gamma }_{ij}$ with ${\gamma }_{ij}^o$ and $\Delta {\gamma }_{ij}$ denoting the nominal part and uncertain part of ${\gamma }_{ij}$, respectively, $j=1,2,3$. Without a loss of generality, suppose ${\gamma }_{i3}^o\ne 0$. For $i=1,\dots ,N$, let $w_i=\mathrm{col}(\Delta {\gamma }_{i1},\Delta {\gamma }_{i2},\Delta {\gamma }_{i3})\in {\mathbb{R}}^3$ denote the vector of uncertain parameters of the ith AUV. Obviously, $w_i=0$ if and only if ${\gamma }_{ij}={\gamma }_{ij}^o$, i.e., there is no system uncertainty. In what follows, let $\mathbb{W}\subseteq {\mathbb{R}}^3$ be a compact set containing the origin of ${\mathbb{R}}^3$.

For $i=1,\dots ,N$, define the state of the ith AUV as $x_i\left(t\right)=\mathrm{col}(p_i\left(t\right),v_i\left(t\right))$. Then, system (1) can be rewritten into the following compact form:

$$\begin{array}{cc}\hfill {\dot{x}}_i& =A_i{x}_i+B_i{u}_i\hfill \end{array}$$

(2a)

$$\begin{array}{cc}\hfill p_i& =C_i{x}_i\hfill \end{array}$$

(2b)

where

$$\begin{array}{cc}{A}_i=\left[\begin{array}{cc}0& I_n\\ {\gamma }_{i1}{I}_n& {\gamma }_{i2}{I}_n\end{array}\right],\hfill & B_i=\left[\begin{array}{c}0\\ {\gamma }_{i3}{I}_n\end{array}\right]\\ C_i=\left[\begin{array}{cc}{I}_n& 0\end{array}\right].\hfill & \end{array}$$

Here, $I_n$ denotes an n-dimensional identity matrix.

Moreover, define the nominal parts of the system matrices $A_i$ and $B_i$ as follows:

$$A_i^o=\left[\begin{array}{cc}0& I_n\\ {\gamma }_{i1}^o{I}_n& {\gamma }_{i2}^o{I}_n\end{array}\right],B_i^o=\left[\begin{array}{c}0\\ {\gamma }_{i3}^o{I}_n\end{array}\right].$$

Since ${\gamma }_{i3}^o\ne 0$, it can be easily verified that $(A_i^o,B_i^o)$ is controllable.

Consider the following polynomial path

$$r_0\left(t\right)=a_m{t}^m+a_{m-1}{t}^{m-1}+\dots +a_{1}t+a_0$$

(3)

where m is some non-negative integer, and $a_i\in {\mathbb{R}}^n$, $i=0,1,\dots ,m$ are constant vectors.

The distributed polynomial path tracking of the swarm system (2) is defined in the following way. For each AUV, denote the local formation command as $r_{fi}\in {\mathbb{R}}^n$, which defines the relative position of the ith AUV with respect to the polynomial path. Then, the absolute trajectory tracking error for the ith AUV is defined by

$$e_i=p_i-r_0-r_{fi}.$$

(4)

Note that an important property of the polynomial path (3) is that it can be generated by a virtual leader system in the following form:

$$\begin{array}{cc}\hfill {\dot{\ell }}_0& =\Xi {\ell }_0\hfill \end{array}$$

(5a)

$$\begin{array}{cc}\hfill r_0& =\Pi {\ell }_0\hfill \end{array}$$

(5b)

where

$$\begin{array}{cc}\hfill \Xi & =I_n\otimes \xi \hfill \\ \hfill \Pi & =I_n\otimes \pi \hfill \end{array}$$

(6)

with

$$\begin{array}{cc}\hfill \xi & =\left[\begin{array}{ccccc}0& 1& 0& \cdots & 0\\ 0& 0& 1& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & 1\\ 0& 0& 0& \cdots & 0\end{array}\right]\in {\mathbb{R}}^{(m+1)\times (m+1)}\hfill \\ \hfill \pi & =\left[\begin{array}{ccccc}1& 0& 0& \cdots & 0\end{array}\right]\in {\mathbb{R}}^{1\times (m+1)}.\hfill \end{array}$$

(7)

Here, ⊗ denotes the Kronecker product of matrices, and ${\ell }_0\in {\mathbb{R}}^{n(m+1)}$ denotes the internal state of the virtual leader. Note that $(\pi ,\xi )$ is observable, and so is $(\Pi ,\Xi )$.

Remark 1.

In this paper, we consider the case where the polynomial path cannot be known in advance by any AUV. Instead, it will be estimated in a distributed way by each AUV depending solely on neighboring information exchange by the distributed observer.

The communication network for the virtual leader system (5) and the swarm system (2) is described by a switching graph ${\overline{\mathcal{G}}}_{\sigma \left(t\right)}=(\overline{\mathcal{V}},{\overline{\mathcal{E}}}_{\sigma \left(t\right)})$ with $\overline{\mathcal{V}}=\{0,1,\dots ,N\}$ and ${\overline{\mathcal{E}}}_{\sigma \left(t\right)}=\{(i,j),i,j\in \mathcal{V},i\ne j\}$. Here, the node 0 is associated with the virtual leader, and the node i, $i=1,\dots ,N$ is associated with the ith AUV. For $i,j=1,\dots ,N$, $i\ne j$, $(i,j)\in {\overline{\mathcal{E}}}_{\sigma \left(t\right)}$ if and only if the jth AUV can receive information from the ith AUV. Moreover, for $i=1,\dots ,N$, $(0,i)\in {\overline{\mathcal{E}}}_{\sigma \left(t\right)}$ if and only if the ith AUV can receive information from the virtual leader. Let the weighted adjacency matrix of the digraph ${\overline{\mathcal{G}}}_{\sigma \left(t\right)}$ be ${\overline{\mathcal{A}}}_{\sigma \left(t\right)}=\left[a_{ij}\left(t\right)\right]\in {\mathbb{R}}^{(N+1)\times (N+1)}$. Define a subgraph ${\mathcal{G}}_{\sigma \left(t\right)}$ of ${\overline{\mathcal{G}}}_{\sigma \left(t\right)}$ as ${\mathcal{G}}_{\sigma \left(t\right)}=(\mathcal{V},{\mathcal{E}}_{\sigma \left(t\right)})$ with $\mathcal{V}=\{1,\dots ,N\}$ and ${\mathcal{E}}_{\sigma \left(t\right)}={\overline{\mathcal{E}}}_{\sigma \left(t\right)}\cap \{\mathcal{V}\times \mathcal{V}\}$. Let ${\mathcal{L}}_{\sigma \left(t\right)}$ be the Laplacian of ${\mathcal{G}}_{\sigma \left(t\right)}$ and $H_{\sigma \left(t\right)}={\mathcal{L}}_{\sigma \left(t\right)}+\mathrm{diag}\{a_{10}\left(t\right),\dots ,a_{N0}\left(t\right)\}$.

The following assumptions are imposed on the communication graphs ${\overline{\mathcal{G}}}_{\sigma \left(t\right)}$ and ${\mathcal{G}}_{\sigma \left(t\right)}$, respectively.

Assumption 1.

There exists a subsequence $\{{\varpi }_k:k=0,1,2,\dots \}$ of $\{k=0,1,2,\dots \}$ satisfying $t_{{\varpi }_{k+1}}-t_{{\varpi }_k}<{\nu }_{\varpi }$ for some ${\nu }_{\varpi }>0$, such that every node i, $i=1,\dots ,N$, is reachable from node 0 in the union graph ${\bigcup }_{r={\varpi }_k}^{{\varpi }_{k+1}-1}{\overline{\mathcal{G}}}_{\sigma \left(t_r\right)}$.

Assumption 2.

The switching graph ${\mathcal{G}}_{\sigma \left(t\right)}$ is undirected. Moreover, there exists a subsequence $\{{\vartheta }_k:k=0,1,2,\dots \}$ of $\{k=0,1,2,\dots \}$ satisfying $t_{{\vartheta }_{k+1}}-t_{{\vartheta }_k}<{\nu }_{\vartheta }$ for some ${\nu }_{\vartheta }>0$, such that the union graph ${\bigcup }_{r={\vartheta }_k}^{{\vartheta }_{k+1}-1}{\mathcal{G}}_{\sigma \left(t_r\right)}$ contains a spanning tree.

Remark 2.

Both Assumptions 1 and 2 are referred to as the jointly connected condition in the literature [32], where Assumption 1 is used for modeling the communication networks for leader–follower-type multi-agent systems, whereas Assumption 2 is used for modeling the communication networks for leaderless-type multi-agent systems. The jointly connected condition is possibly the mildest condition ever imposed on communication networks, and can tolerate the extreme case where the communication network topology is disconnected for all of the time instants.

Now, we are ready to formulate the polynomial path tracking problem as follows.

Problem 1.

Given systems (2), (5) and the communication graph ${\overline{\mathcal{G}}}_{\sigma \left(t\right)}$, design a distributed control law $u_i$ in the following form:

$$\begin{array}{cc}\hfill u_i& =f_i(v_i,{\varsigma }_i,{\varsigma }_j,p_j-p_i,j\in {\mathcal{N}}_i\left(t\right))\hfill \end{array}$$

(8a)

$$\begin{array}{cc}\hfill {\dot{\varsigma }}_i& =g_i(v_i,{\varsigma }_i,{\varsigma }_j,p_j-p_i,j\in {\mathcal{N}}_i\left(t\right))\hfill \end{array}$$

(8b)

such that there exists $\mathbb{W}\subseteq {\mathbb{R}}^3$ for any $w_i\in \mathbb{W}$, and for any system initial condition, there exists some constant vector $e_c\in {\mathbb{R}}^3$, known or not, satisfying

$$\underset{t\to \infty }{lim}{e}_i\left(t\right)=e_c,i=1,\dots ,N.$$

(9)

Remark 3.

In this paper, it is assumed that the absolute positions $p_i$s of the AUVs are not available. Therefore, it is, in general, impossible to drive $e_i$ to zero. While, in practice, the absolute position tracking is usually immaterial, it is the formation generation and keeping that matters. Thus, instead of regulating $e_i$ to zero, it suffices for the absolute tracking errors to converge to a common constant vector, which, in turn, implies the achievement of formation generation and keeping.

4. Main Results

The distributed robust control scheme proposed in this paper is composed of three parts. First, an output-based distributed observer was invoked to recover the polynomial path for each AUV. Second, a pseudo position estimator was adopted to generate the pseudo position for each AUV. Finally, based on the estimated polynomial path and the pseudo position, a certainty equivalent robust internal model control law was synthesized to achieve reference trajectory tracking. The details of these three parts are presented in sequence as follows.

First, we introduce the output-based distributed observer for the virtual leader.

Since $(\pi ,\xi )$ is observable, let $\chi \in {\mathbb{R}}^{(m+1)\times (m+1)}$ be the positive definite matrix solution to the following algebraic Riccati equation

$$\chi {\xi }^T+\xi \chi -\chi {\pi }^T\pi \chi +I_{m+1}=0.$$

(10)

Let

$$L=I_n\otimes \left(\chi {\pi }^T\right).$$

(11)

Then, for $i=1,\dots ,N$, design

$$\begin{array}{cc}\hfill {\dot{\ell }}_i& =\Xi {\ell }_i+{\mu }_{\ell }L\sum _{i=0}^N{a}_{ij}\left(t\right)(r_j-r_i)\hfill \end{array}$$

(12a)

$$\begin{array}{cc}\hfill r_i& =\Pi {\ell }_i\hfill \end{array}$$

(12b)

where ${\ell }_i$ and $r_i$ are the estimates of ${\ell }_0$ and $r_0$, respectively, and ${\mu }_{\ell }>0$ is the observer gain.

For $i=1,\dots ,N$, define ${\tilde{\ell }}_i={\ell }_i-{\ell }_0$ and ${\tilde{r}}_i=r_i-r_0$. Then,

$$\begin{array}{cc}\hfill {\dot{\tilde{\ell }}}_i& =\Xi {\tilde{\ell }}_i+{\mu }_{\ell }L\sum _{i=0}^N{a}_{ij}\left(t\right)({\tilde{r}}_j-{\tilde{r}}_i)\hfill \\ \hfill & =\Xi {\tilde{\ell }}_i+{\mu }_{\ell }L\Pi \sum _{i=0}^N{a}_{ij}\left(t\right)({\tilde{\ell }}_j-{\tilde{\ell }}_i).\hfill \end{array}$$

(13)

For $x_i\in {\mathbb{R}}^{n_i}$, $i=1,\dots ,N$, define the notation $\mathrm{col}(x_1,\dots ,x_N)={[x_1^T,\dots ,x_N^T]}^T$. Let $\tilde{\ell }=\mathrm{col}({\tilde{\ell }}_1,\dots ,{\tilde{\ell }}_N)$. Then, we have

$$\dot{\tilde{\ell }}=(I_N\otimes \Xi -H_{\sigma \left(t\right)}\otimes ({\mu }_{\ell }L\Pi ))\tilde{\ell }.$$

(14)

Using Theorem 4.1 of [32], we have the following result.

Lemma 1.

Given systems (5) and (12), under Assumptions 1 and 2, for any system initial condition, and any ${\mu }_{\ell }>0$, both ${\tilde{\ell }}_i\left(t\right)$ and ${\tilde{r}}_i\left(t\right)$ tend to zero exponentially as $t\to \infty$.

Remark 4.

The dynamic compensator (12) is called an output-based distributed observer of the virtual leader system (5) in the sense that: (1) it relies solely on the output of the virtual leader $r_0$; (2) it only requires local information of neighboring AUVs; (3) it can recover the information of the virtual leader for each AUV. In this way, the information of the virtual leader is transmitted to each AUV over the unreliable jointly connected switching network ${\overline{\mathcal{G}}}_{\sigma \left(t\right)}$.

Next, we introduce the pseudo position estimator. For $i=1,\dots ,N$, design

$$\begin{array}{cc}\hfill {\dot{\wp }}_i& =v_i+{\mu }_{\wp }\sum _{j=1}^N{a}_{ij}\left(t\right)({\wp }_j-{\wp }_i-p_{ji})\hfill \end{array}$$

(15a)

$$\begin{array}{cc}\hfill p_{ji}& =p_j-p_i\hfill \end{array}$$

(15b)

where ${\mu }_{\wp }>0$ is the consensus gain.

Define ${\tilde{\wp }}_i={\wp }_i-p_i$. Then, we have

$$\begin{array}{cc}\hfill {\dot{\tilde{\wp }}}_i& =v_i-v_i+{\mu }_{\wp }\sum _{j=1}^N{a}_{ij}\left(t\right)({\wp }_j-{\wp }_i-(p_j-p_i))\hfill \\ \hfill & ={\mu }_{\wp }\sum _{j=1}^N{a}_{ij}\left(t\right)({\tilde{\wp }}_j-{\tilde{\wp }}_i).\hfill \end{array}$$

(16)

Then, under Assumption 1, using Theorem 2.8 of [1], it follows that, for any ${\mu }_{\wp }>0$,

$$\underset{t\to \infty }{lim}{\tilde{\wp }}_i\left(t\right)={\wp }_c$$

(17)

exponentially for some static vector ${\wp }_c\in {\mathbb{R}}^3$.

Remark 5.

For $i=1,\dots ,N$, define

$${\check{p}}_i=p_i+{\wp }_c$$

and

$${\tilde{p}}_i={\wp }_i-{\check{p}}_i.$$

Then, using (17), it follows that

$$\underset{t\to \infty }{lim}{\tilde{p}}_i\left(t\right)=0$$

(18)

exponentially. Equation (18) means that, though the absolute position $p_i$ is not available, it is possible to generate a pseudo position ${\wp }_i$ for each AUV ensuring that the differences between the pseudo positions and the absolute positions of all of the AUVs are a common constant vector. It is this very result that leads to the solution to the polynomial path formation problem by a distributed control law in the form of (8).

In terms of ${\check{p}}_i$, the system dynamics (1) become

$${\dot{\check{p}}}_i=v_i$$

(19a)

$$\begin{array}{cc}\hfill {\dot{v}}_i& ={\gamma }_{i1}({\check{p}}_i-{\wp }_c)+{\gamma }_{i2}{v}_i+{\gamma }_{i3}{u}_i\hfill \\ \hfill & ={\gamma }_{i1}{\check{p}}_i+{\gamma }_{i2}{v}_i+{\gamma }_{i3}{u}_i-{\gamma }_{i1}{\wp }_c.\hfill \end{array}$$

(19b)

Moreover, we define the new tracking error as

$$\begin{array}{cc}\hfill {\check{e}}_i& ={\check{p}}_i-r_0-r_{fi}\hfill \\ \hfill & =p_i+{\wp }_c-r_0-r_{fi}\hfill \\ \hfill & =e_i+{\wp }_c.\hfill \end{array}$$

(20)

Then, ${\check{e}}_i=0$ if and only if $e_i=-{\wp }_c$. As a result, to solve Problem 1, it suffices to solve the following problem.

Problem 2.

Given systems (5), (19) and the communication graph ${\overline{\mathcal{G}}}_{\sigma \left(t\right)}$, design a distributed control law $u_i$ in the form of (8) such that there exists $\mathbb{W}\subseteq {\mathbb{R}}^3$ for any $w_i\in \mathbb{W}$, and, for any system initial condition,

$$\underset{t\to \infty }{lim}{\check{e}}_i\left(t\right)=0,i=1,\dots ,N.$$

(21)

To solve Problem 2, we will rewrite the system dynamics in a new compact form. First, define the following augmented exosystem:

$$\dot{v}=\left[\begin{array}{c}0\\ & \Xi \end{array}\right]v\triangleq \overline{\Xi }v,v\left(0\right)=\left[\begin{array}{c}1\\ {\ell }_0\left(0\right)\end{array}\right].$$

(22)

Therefore, $v\left(t\right)=\mathrm{col}(1,{\ell }_0\left(t\right))$. Then, letting ${\check{x}}_i=\mathrm{col}({\check{p}}_i,v_i)$ gives

$$\begin{array}{cc}\hfill {\dot{\check{x}}}_i& =A_i{\check{x}}_i+B_i{u}_i+E_{i}v\hfill \end{array}$$

(23a)

$$\begin{array}{cc}\hfill {\check{e}}_i& =C_i{\check{x}}_i+F_{i}v\hfill \end{array}$$

(23b)

where

$$\begin{array}{cc}\hfill E_i& =\left[\begin{array}{cc}{0}_{n\times 1}& 0_{n\times \left(n\right(m+1\left)\right)}\\ -{\gamma }_{i1}{\wp }_c& 0_{n\times \left(n\right(m+1\left)\right)}\end{array}\right]\hfill \\ \hfill F_i& =\left[\begin{array}{cc}-r_{fi}& -\Pi \end{array}\right].\hfill \end{array}$$

Note that the minimal polynomials of $\xi$, $\Xi$ and $\overline{\Xi }$ are the same. Therefore, let

$${\mathfrak{G}}_1=\xi ,{\mathfrak{G}}_2=\left[\begin{array}{c}0\\ ⋮\\ 0\\ 1\end{array}\right]\in {\mathbb{R}}^{m+1}$$

(24)

and thus the following matrix pair

$$G_1=I_n\otimes {\mathfrak{G}}_1,G_2=I_n\otimes {\mathfrak{G}}_2$$

(25)

incorporates an n-copy internal model of $\overline{\Xi }$.

Since all of the eigenvalues of $\overline{\Xi }$ are zero, for any eigenvalue $\lambda$ of $\overline{\Xi }$, note that ${\gamma }_{i3}^o\ne 0$ gives

$$\begin{array}{cc}\hfill & \mathrm{rank}\left[\begin{array}{cc}{A}_i^o-\lambda I_n& B_i^o\\ C_i& 0\end{array}\right]\hfill \\ \hfill =& \mathrm{rank}\left[\begin{array}{ccc}0& I_n& 0\\ {\gamma }_{i1}^o{I}_n& {\gamma }_{i2}^o{I}_n& {\gamma }_{i3}^o{I}_n\\ I_n& 0& 0\end{array}\right]\hfill \\ \hfill =& 3n.\hfill \end{array}$$

Thus, by Lemma 1.26 of [39], the matrix pair $({\overline{A}}_i^o,{\overline{B}}_i^o)$ is able to be stabilized, where

$${\overline{A}}_i^o=\left[\begin{array}{cc}{A}_i^o& 0\\ G_2{C}_i& G_1\end{array}\right],{\overline{B}}_i^o=\left[\begin{array}{c}{B}_i^o\\ 0\end{array}\right]$$

(26)

Furthermore, let

$$K_i=\left[\begin{array}{cc}{K}_{i1}& K_{i2}\end{array}\right]$$

be such that

$${\overline{A}}_i^o+{\overline{B}}_i^o{K}_i=\left[\begin{array}{cc}{A}_i^o+B_i^o{K}_{i1}& B_i^o{K}_{i2}\\ G_2{C}_i& G_1\end{array}\right]$$

is Hurwitz.

For $i=1,\dots ,N$, let ${\widehat{x}}_i=\mathrm{col}({\wp }_i,v_i)$ and ${\tilde{x}}_i={\widehat{x}}_i-{\check{x}}_i=\mathrm{col}({\tilde{p}}_i,0_{n\times 1})$. Therefore, ${lim}_{t\to \infty }{\tilde{x}}_i\left(t\right)=0$ exponentially using (18). Now, we are ready to present the certainty equivalent robust internal model control law as follows:

$$\begin{array}{cc}\hfill u_i& =K_{i1}{\widehat{x}}_i+K_{i2}{z}_i\hfill \end{array}$$

(27a)

$$\begin{array}{cc}\hfill {\dot{z}}_i& =G_1{z}_i+G_2{\widehat{e}}_i\hfill \end{array}$$

(27b)

$$\begin{array}{cc}\hfill {\widehat{e}}_i& ={\wp }_i-r_i-r_{fi}.\hfill \end{array}$$

(27c)

The overall distributed robust control scheme proposed in this paper is composed of the distributed observer (12), the pseudo position estimator (15) and the certainty equivalent robust internal model control law (27). The information flow among different parts of the distributed robust control scheme is illustrated in Figure 1. A pseudo code used to calculate the control input by the distributed robust control scheme for the ith AUV, $i=1,\dots ,N$, is given by Algorithm 1.

Algorithm 1 Calculating the control input by the distributed robust control scheme for the ith AUV

Input:$r_j$, $p_j-p_i$, $j\in {\mathcal{N}}_i\left(t\right)$ Output:$u_i$

1: According to the dimension of the position output n and the order of the polynomial path m, determine the matrix pairs (7) and (6).

2: Solve the Riccati Equation (10) and design L using (11).

3: Select any ${\mu }_{\ell }>0$ and implement the distributed observer (12).

4: Select any ${\mu }_{\wp }>0$ and implement the pseudo position estimator (15) based on $v_i$ and $p_j-p_i$.

5: Determine $({\mathfrak{G}}_1,{\mathfrak{G}}_2)$ using (24), $(G_1,G_2)$ using (25) and $({\overline{A}}_i^o,{\overline{B}}_i^o)$ using (26).

6: Select $K_i$ such that ${\overline{A}}_i^o+{\overline{B}}_i^o{K}_i$ is Hurwitz.

7: Based on $r_i$ from the distributed observer (12) and ${\wp }_i$ from the pseudo position estimator (15), implement the certainty equivalent robust internal model control law (27).

The main result of this paper is presented as follows.

Theorem 1.

Given systems (5), (23) and the communication graph ${\overline{\mathcal{G}}}_{\sigma \left(t\right)}$, under Assumptions 1 and 2, Problem 2 is solvable by the control law composed of (12), (15) and (27) for any ${\mu }_{\ell },{\mu }_{\wp }>0$.

Proof. 

Define, for $i=1,\dots ,N$,

$${\overline{A}}_i^o=\left[\begin{array}{cc}{A}_i& 0\\ G_2{C}_i& G_1\end{array}\right],{\overline{B}}_i^o=\left[\begin{array}{c}{B}_i\\ 0\end{array}\right].$$

Since ${\overline{A}}_i^o+{\overline{B}}_i^o{K}_i$ is Hurwits, there exists $\mathbb{W}\subseteq {\mathbb{R}}^3$, containing the origin of ${\mathbb{R}}^3$ such that

$${\overline{A}}_i+{\overline{B}}_i{K}_i=\left[\begin{array}{cc}{A}_i+B_i{K}_{i1}& B_i{K}_{i2}\\ G_2{C}_i& G_1\end{array}\right]$$

is Hurwitz for any $w_i\in \mathbb{W}$. Then, using Lemma 1.27 of [39], the following matrix equation

$$\begin{array}{cc}\hfill X_i\overline{\Xi }& =(A_i+B_i{K}_{i1})X_i+B_i{K}_{i2}{Z}_i+E_i\hfill \end{array}$$

(28a)

$$\begin{array}{cc}\hfill Z_i\overline{\Xi }& =G_1{Z}_i+G_2(C_i{X}_i+F_i)\hfill \end{array}$$

(28b)

have a unique solution pair $(X_i,Z_i)$, which, in addition, satisfies

$$0=C_i{X}_i+F_i.$$

(29)

Substituting (27) into (23) gives

$$\begin{array}{cc}\hfill {\dot{\check{x}}}_i& =A_i{\check{x}}_i+B_i{u}_i+E_{i}v\hfill \\ \hfill & =A_i{\check{x}}_i+B_i{K}_{i1}{\widehat{x}}_i+B_i{K}_{i2}{z}_i+E_{i}v\hfill \\ \hfill & =A_i{\check{x}}_i+B_i{K}_{i1}{\check{x}}_i+B_i{K}_{i2}{z}_i+E_{i}v+B_i{K}_{i1}{\tilde{x}}_i\hfill \\ \hfill & =(A_i+B_i{K}_{i1}){\check{x}}_i+B_i{K}_{i2}{z}_i+E_{i}v+B_i{K}_{i1}{\tilde{x}}_i\hfill \end{array}$$

(30)

and

$$\begin{array}{cc}\hfill {\dot{z}}_i& =G_1{z}_i+G_2{\widehat{e}}_i\hfill \\ \hfill & =G_1{z}_i+G_2({\check{e}}_i+r_0-r_i+{\wp }_i-{\check{p}}_i)\hfill \\ \hfill & =G_1{z}_i+G_2{\check{e}}_i+G_2{\tilde{p}}_i-G_2{\tilde{r}}_i.\hfill \end{array}$$

(31)

Next, for $i=1,\dots ,N$, define

$$\begin{array}{cc}\hfill {\overline{x}}_i& ={\check{x}}_i-X_{i}v\hfill \\ \hfill {\overline{z}}_i& =z_i-Z_{i}v.\hfill \end{array}$$

Then, using (28), we have

$$\begin{array}{cc}\hfill {\dot{\overline{x}}}_i& =(A_i+B_i{K}_{i1}){\check{x}}_i+B_i{K}_{i2}{z}_i+E_{i}v+B_i{K}_{i1}{\tilde{x}}_i-X_i\overline{\Xi }v\hfill \\ \hfill & =(A_i+B_i{K}_{i1})({\overline{x}}_i+X_{i}v)+B_i{K}_{i2}{Z}_{i}v+B_i{K}_{i2}{\overline{z}}_i+E_{i}v-X_i\Xi v+B_i{K}_{i1}{\tilde{x}}_i\hfill \\ \hfill & =(A_i+B_i{K}_{i1}){\overline{x}}_i+B_i{K}_{i2}{\overline{z}}_i+B_i{K}_{i1}{\tilde{x}}_i\hfill \end{array}$$

(32)

and

$$\begin{array}{cc}\hfill {\dot{\overline{z}}}_i& =G_1{z}_i+G_2{\check{e}}_i+G_2{\tilde{p}}_i-G_2{\tilde{r}}_i-Z_i\overline{\Xi }v\hfill \\ \hfill & =G_1({\overline{z}}_i+Z_{i}v)+G_2(C_i{\check{x}}_i+F_{i}v)+G_2{\tilde{p}}_i-G_2{\tilde{r}}_i-Z_i\overline{\Xi }v\hfill \\ \hfill & =G_1({\overline{z}}_i+Z_{i}v)+G_2(C_i({\overline{x}}_i+X_{i}v)+F_{i}v)+G_2{\tilde{p}}_i-G_2{\tilde{r}}_i-Z_i\overline{\Xi }v\hfill \\ \hfill & =G_2{C}_i{\overline{x}}_i+G_1{\overline{z}}_i+G_2{\tilde{p}}_i-G_2{\tilde{r}}_i.\hfill \end{array}$$

(33)

For $i=1,\dots ,N$, define ${\rho }_i=\mathrm{col}({\overline{x}}_i,{\overline{z}}_i)$. Then, it follows that

$$\begin{array}{c}\hfill {\dot{\rho }}_i={\mathsf{{\rm Y}}}_i{\rho }_i+{\psi }_i\end{array}$$

(34)

with

$$\begin{array}{cc}\hfill {\mathsf{{\rm Y}}}_i& =\left[\begin{array}{cc}{A}_i+B_i{K}_{i1}& B_i{K}_{i2}\\ G_2{C}_i& G_1\end{array}\right]\hfill \\ \hfill {\psi }_i& =\left[\begin{array}{c}{B}_i{K}_{i1}{\tilde{x}}_i\\ G_2({\tilde{p}}_i-{\tilde{r}}_i)\end{array}\right].\hfill \end{array}$$

(35)

Since ${\mathsf{{\rm Y}}}_i$ is Hurwitz for any $w_i\in \mathbb{W}$, and ${\tilde{x}}_i\left(t\right),{\psi }_i\left(t\right)$ decay to zero exponentially as $t\to \infty$, using Lemma 2.5 of [32], it follows that

$$\underset{t\to \infty }{lim}{\rho }_i\left(t\right)=0.$$

(36)

Moreover, using (29), it follows that

$$\begin{array}{cc}\hfill {\check{e}}_i& =C_i{\check{x}}_i+F_{i}v\hfill \\ \hfill & =C_i({\overline{x}}_i+X_{i}v)+F_{i}v\hfill \\ \hfill & =C_i{\overline{x}}_i.\hfill \end{array}$$

(37)

Therefore, using (36), ${lim}_{t\to \infty }{\check{e}}_i\left(t\right)=0$ and thus the proof is complete. □

5. Numerical Simulations

In this section, we will use numerical simulations to illustrate and validate the proposed control scheme. The control task is taken from [38], where a fleet of AUVs executed the formation task in Monterey Bay during August 2003 to observe and predict ocean processes.

5.1. Aim of the Experiment

We begin with the aim of the experiment. In [38], the aim of the experiment is to drive the AUV fleet to cruise in the ocean in a static triangular formation while following the reference trajectory of a straight line, illustrated by Figure 2.

In this paper, we consider a similar task as in [38]. Consider a swarm system of four AUVs, whose dynamics are given by

$$\begin{array}{cc}\hfill {\dot{p}}_i& =v_i\hfill \end{array}$$

(38a)

$$\begin{array}{cc}\hfill {\dot{v}}_i& ={\gamma }_{i1}{p}_i+{\gamma }_{i2}{v}_i+{\gamma }_{i3}{u}_i\hfill \end{array}$$

(38b)

where $p_i,v_i,u_i\in {\mathbb{R}}^3$. For $i=1,\dots ,4$, suppose the system parameters are given by ${\gamma }_{i1}^o=5$, ${\gamma }_{i2}^o=-10$, ${\gamma }_{i3}^o=8$, $\Delta {\gamma }_{i1}\in [-1,1]$, $\Delta {\gamma }_{i2}\in [-2,2]$, $\Delta {\gamma }_{i3}\in [-1,1]$.

The polynomial path is given by

$$r_0\left(t\right)=\left[\begin{array}{c}2t-10\\ -4t+5\\ -t\end{array}\right].$$

The local formation vector is given by

$$r_{f1}=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right],r_{f2}=\left[\begin{array}{c}0\\ 0\\ 5\end{array}\right],r_{f3}=\left[\begin{array}{c}5\\ 0\\ -3\end{array}\right],r_{f4}=\left[\begin{array}{c}-5\\ 0\\ -3\end{array}\right].$$

The communication network is shown by Figure 3. In particular, the communication network ${\overline{\mathcal{G}}}_{\sigma \left(t\right)}$ is assumed to switch among six subgraphs ${\overline{\mathcal{G}}}_1$, ⋯ ${\overline{\mathcal{G}}}_6$, periodically every $T_c$ sec. Suppose that $T_c=0.1$. It can be verified that Assumptions 1 and 2 are both satisfied. The distinguished characteristic of the communication network ${\overline{\mathcal{G}}}_{\sigma \left(t\right)}$ is that it is disconnected the entire time.

5.2. Methodology

Next, we will follow Algorithm 1 to conceive the distributed robust control scheme.

1.

The dimension of the position output is 3, and the order of the polynomial path is 1. Therefore, we have

$$\xi =\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right],\pi =\left[\begin{array}{cc}1& 0\end{array}\right]$$

(39)

and

$$\begin{array}{c}\hfill \Xi =I_3\otimes \xi ,\Pi =I_3\otimes \pi .\end{array}$$

(40)

2.

The solution to the following Riccati equation

$$\chi {\xi }^T+\xi \chi -\chi {\pi }^T\pi \chi +I_2=0$$

(41)

is

$$\chi =\left[\begin{array}{cc}1.7321& 1\\ 1& 1.7321\end{array}\right]$$

(42)

and thus

$$L=I_3\otimes \left(\chi {\pi }^T\right)=I_3\otimes \left[\begin{array}{c}1.7321\\ 1\end{array}\right].$$

(43)

3.

Select ${\mu }_{\ell }=10$ and design the distributed observer as follows

$$\begin{array}{cc}\hfill {\dot{\ell }}_i& =\Xi {\ell }_i+{\mu }_{\ell }L\sum _{i=0}^N{a}_{ij}\left(t\right)(r_j-r_i)\hfill \end{array}$$

(44a)

$$\begin{array}{cc}\hfill r_i& =\Pi {\ell }_i\hfill \end{array}$$

(44b)

where ${\ell }_i\in {\mathbb{R}}^6$.

4.

Select ${\mu }_{\wp }=10$ and design the pseudo position estimator as follows

$$\begin{array}{cc}\hfill {\dot{\wp }}_i& =v_i+{\mu }_{\wp }\sum _{j=1}^N{a}_{ij}\left(t\right)({\wp }_j-{\wp }_i-p_{ji})\hfill \end{array}$$

(45a)

$$\begin{array}{cc}\hfill p_{ji}& =p_j-p_i\hfill \end{array}$$

(45b)

where ${\wp }_i\in {\mathbb{R}}^3$.

5.

Using (24)

$${\mathfrak{G}}_1=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right],{\mathfrak{G}}_2=\left[\begin{array}{c}0\\ 1\end{array}\right]$$

(46)

and, using (25),

$$G_1=I_3\otimes {\mathfrak{G}}_1,G_2=I_3\otimes {\mathfrak{G}}_2.$$

(47)

Using (38), it follows that

$$A_i^o=\left[\begin{array}{cc}{0}_{3\times 3}& I_3\\ 5I_3& -10I_3\end{array}\right],B_i^o=\left[\begin{array}{c}{0}_{3\times 3}\\ 8I_3\end{array}\right],C_i=I_3\otimes \left[\begin{array}{cc}1& 0\end{array}\right]$$

(48)

and, thus, using (26),

$${\overline{A}}_i^o=\left[\begin{array}{cc}{A}_i^o& 0_{6\times 6}\\ G_2{C}_i& G_1\end{array}\right]=\left[\begin{array}{cc}{A}_i^o& 0_{6\times 6}\\ I_3\otimes \left[\begin{array}{cc}0& 0\\ 1& 0\end{array}\right]& G_1\end{array}\right],{\overline{B}}_i^o=\left[\begin{array}{c}{B}_i^o\\ 0_{6\times 3}\end{array}\right].$$

(49)

6.

By letting

$$K_i=\left[\begin{array}{cc}{K}_{i1}& K_{i2}\end{array}\right]$$

with

$$K_{i1}=\left[\begin{array}{cccccc}-39.6591& -0.9323& 0.4840& -2.3884& -0.0400& 0.0213\\ -1.1874& -48.0646& -3.1420& -0.0519& -2.7540& -0.1312\\ 0.8779& -3.1444& -47.5545& 0.0378& -0.1308& -2.7325\end{array}\right]$$

and

$$K_{i2}=\left[\begin{array}{cccccc}-314.8926& -182.7741& -17.9939& -7.1494& 8.7643& 3.6004\\ -21.7605& -8.8925& -472.3295& -246.2606& -63.8461& -24.72\\ 16.8854& 6.7136& -64.4250& -24.8377& -461.9890& -242.2642\end{array}\right]$$

it follows that the eigenvalues of ${\overline{A}}_i^o+{\overline{B}}_i^o{K}_i$ are located at

$$\{-5,-5.5,-6,-6.5,-7,-7.5,-8,-8.5,-9,-9.5,-10,-10.5\}$$

that is, ${\overline{A}}_i^o+{\overline{B}}_i^o{K}_i$ is Hurwitz.

7.

Let ${\widehat{x}}_i=\mathrm{col}({\wp }_i,v_i)$ and the certainty equivalent robust internal model control law be designed as follows:

$$\begin{array}{cc}\hfill u_i& =K_{i1}{\widehat{x}}_i+K_{i2}{z}_i\hfill \end{array}$$

(50a)

$$\begin{array}{cc}\hfill {\dot{z}}_i& =G_1{z}_i+G_2{\widehat{e}}_i\hfill \end{array}$$

(50b)

$$\begin{array}{cc}\hfill {\widehat{e}}_i& ={\wp }_i-r_i-r_{fi}.\hfill \end{array}$$

(50c)

5.3. Results

Now, we examine the system performance using simulation results. Suppose that the initial positions and velocities of the AUVs are given by

$$\begin{array}{cccc}{p}_1\left(0\right)=\left[\begin{array}{c}5\\ 5\\ 0\end{array}\right],& p_2\left(0\right)=\left[\begin{array}{c}5\\ -5\\ 0\end{array}\right],& p_3\left(0\right)=\left[\begin{array}{c}-5\\ 5\\ 0\end{array}\right],& p_4\left(0\right)=\left[\begin{array}{c}-5\\ -5\\ 0\end{array}\right]\\ v_1\left(0\right)=\left[\begin{array}{c}0\\ 0.6\\ -0.6\end{array}\right],& v_2\left(0\right)=\left[\begin{array}{c}0.5\\ 0\\ -0.2\end{array}\right],& v_3\left(0\right)=\left[\begin{array}{c}0.2\\ 0.3\\ -0.1\end{array}\right],& v_4\left(0\right)=\left[\begin{array}{c}-0.1\\ -0.5\\ -0.4\end{array}\right].\end{array}$$

The initial values of the control laws, i.e., the components of ${\wp }_i\left(0\right)$, ${\ell }_i\left(0\right)$ and $z_i\left(0\right)$, take random values from the interval $[0,0.5]$.

5.3.1. Standard Case

The simulation results are shown in Figure 4, Figure 5, Figure 6 and Figure 7. In particular, the performance of the distributed observer over the unreliable switching communication network ${\overline{\mathcal{G}}}_{\sigma \left(t\right)}$ is shown in Figure 4. It can be seen that the polynomial trajectory has been successfully recovered by the distributed observer of each AUV. The performance of the pseudo position estimator is shown in Figure 5. As proved, differences between the actual positions and the pseudo positions of all of the AUVs will converge to a common constant vector, which, in our case, is very close to zero. The absolute tracking errors for all of the AUVs are shown in Figure 6. It can be seen that these tracking errors also converge to a common constant vector, as required by the control objective (9), i.e., the distributed tracking problem has been successfully achieved by the proposed distributed robust control scheme. Finally, the 3D trajectories of all of the AUVs are plotted in Figure 7, where the process of formation generation and keeping can be seen straightforwardly.

5.3.2. Comparative Study

In this case, we compare the proposed distributed robust control scheme with a typical existing work [29] from the perspective of a communication network exclusively. For simplicity, for the control scheme proposed in [29], it is assumed that the absolute position of the AUV is available for control feedback, and the system parameters are fully known, while keeping in mind that, for the control scheme proposed in this paper, the absolute position of the AUV is not available, and the system parameters are unknown. Suppose that the polynomial path considered in this case is given by

$$r_0\left(t\right)=\left[\begin{array}{c}{t}^2+2t-10\\ 2t^2-4t+5\\ -t\end{array}\right].$$

(51)

First, we consider the ideal static and connected communication network for the result in [29]. Suppose the communication network is the union of the six subgraphs of Figure 3, which is shown in Figure 8. Under the communication graph $\overline{\mathcal{G}}$, the simulation results using the control method in [29] are shown in Figure 9. It can be seen that the tracking errors have been driven to zero asymptotically. Under the communication graph ${\overline{\mathcal{G}}}_{\sigma \left(t\right)}$ defined by Figure 3 with different switching period $T_c$, the simulation results using the control method in [29] are shown in Figure 10 and Figure 11. When the communication network becomes unreliable and switched, the tracking errors will no longer converge to zero, which concludes that the control method in [29] is effective for a static and connected communication network, but cannot effectively deal with an unreliable jointly connected switching communication network. Similar to the design process as in the standard case, we can design the distributed robust control scheme proposed in this paper for the new $r_0$ given by (51). The simulation results using the proposed control scheme of this paper are given in Figure 12 and Figure 13, which show that successful tracking has been achieved for both cases.

6. Methodology Discussion

In this section, we will further examine the effectiveness of the proposed control scheme from the perspective of measurement noise. There are two sources of measurement noises associated with the distributed robust control scheme proposed in this paper.

The first one is the velocity measurement noise imposed on $v_i$. Suppose that

$$v_i^m=v_i+n_{vi}$$

(52)

where $v_i^m,v_i,n_{vi}$ denote the measured velocity, true velocity and velocity measurement noise, respectively. In what follows, we will show, using simulation results, the effect of $n_{vi}$ on the system performance. In the simulations, suppose that the entries of $n_{vi}$ take random values uniformly from $[-n_V,n_V]$, with $n_V>0$ being the magnitude of the velocity measurement noise. Note that the velocity measurement noise will affect the pseudo position estimator (15) and the robust internal model control law (27). Simulation results with $n_V=1$ are shown in Figure 14 and Figure 15, from which, it can be observed that, due to the velocity measurement noise, the differences between the pseudo positions and the actual positions of the AUVs will approximately converge to some common vector, while this common vector is not constant but time-varying. As a result, the steady state trajectory tracking errors of all of the AUVs also approximately converge to some time-varying common vector.

The second one is the relative position measurement noise imposed on $p_j-p_i$. In this scenario, suppose the pseudo position estimator takes the following form

$$\begin{array}{cc}\hfill {\dot{\wp }}_i& =v_i+{\mu }_{\wp }\sum _{j=1}^N{a}_{ij}\left(t\right)({\wp }_j-{\wp }_i-p_{ji})+{\mu }_{\wp }{n}_{\wp i}\hfill \end{array}$$

(53a)

$$\begin{array}{cc}\hfill p_{ji}& =p_j-p_i\hfill \end{array}$$

(53b)

where $n_{\wp i}$ denotes the lumped relative position measurement noise for the ith AUV. Again, we will show, using simulation results, the effect of $n_{\wp i}$ on the system performance. Similarly, suppose that the entries of $n_{\wp i}$ take random values uniformly from $[-n_P,n_P]$, with $n_P>0$ being the magnitude of the relative position measurement noise. Note that the relative position measurement noise will affect the pseudo position estimator (15), and thus the tracking performance of the system. Simulation results with $n_P=0.1$ are shown in Figure 16 and Figure 17. Similar to the case of velocity measurement noise, the differences between the pseudo positions and the actual positions of the AUVs will approximately converge to some time-varying common vector, which, in turn, makes the approximate common path tracking error time-varying too. Note that the gain for the distributed observer ${\mu }_{\wp }$ will amplify the magnitude of the relative position measurement noise. As a result, ${\mu }_{\wp }$ should not be selected as overly large in the presence of relative position measurement noise.

7. Conclusions

In this paper, a distributed robust control scheme is proposed to solve the polynomial path tracking problem for a swarm of uncertain AUVs facing three application challenges. First, the communication network is unreliable, satisfying merely the jointly connected condition. Second, only the relative position measurement between neighboring AUVs over the communication network is available for control feedback. Third, the second-order model dynamics for the AUV contain uncertain system parameters. To address these issues, three control parts were designed constituting the distributed robust control scheme, namely, the distributed observer, the pseudo position estimator and the certainty equivalent robust internal model control law. Comprehensive simulation results have validated the effectiveness of the proposed control scheme, especially the robustness against the unreliable and switching communication network when comparing with other existing results. Moreover, a further discussion on the effect of measurement noises on the system performance was conducted, where it was shown using simulation results that the proposed control scheme shows a certain resiliency with respect to velocity and relative position measurement noises. In this paper, it is assumed that the AUVs are fully actuated, which results in a linear system dynamic model. In the future, it would be interesting to further consider the case of underactuated AUVs with complex nonlinear system dynamics.